Multicolor Turán numbers II -- a generalization of the Ruzsa-Szemerédi theorem and new results on cliques and odd cycles
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arXiv
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| Natura: | Preprint |
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2022
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| _version_ | 1866911819647418368 |
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| author | Kovács, Benedek Nagy, Zoltán Lóránt |
| author_facet | Kovács, Benedek Nagy, Zoltán Lóránt |
| contents | In this paper we continue the study of a natural generalization of Turán's forbidden subgraph problem and the Ruzsa-Szemerédi problem. Let $ex_F(n,G)$ denote the maximum number of edge-disjoint copies of a fixed simple graph $F$ that can be placed on an $n$-vertex ground set without forming a subgraph $G$ whose edges are from different $F$-copies. The case when both $F$ and $G$ are triangles essentially gives back the theorem of Ruzsa and Szemerédi. We extend their results to the case when $F$ and $G$ are arbitrary cliques by applying a number theoretic result due to Erdős, Frankl and Rödl. This extension in turn decides the order of magnitude for a large family of graph pairs, which will be subquadratic, but almost quadratic. Since the linear $r$-uniform hypergraph Turán problems to determine $ex_r^{lin}(n,G)$ form a class of the multicolor Turán problem, following the identity $ex_r^{lin}(n,G)=ex_{K_r}(n,G)$, our results determine the linear hypergraph Turán numbers of every graph of girth $3$ and for every $r$ up to a subpolynomial factor. Furthermore, when $G$ is a triangle, we settle the case $F=C_5$ and give bounds for the cases $F=C_{2k+1}$, $k\ge 3$ as well. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2207_14572 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Multicolor Turán numbers II -- a generalization of the Ruzsa-Szemerédi theorem and new results on cliques and odd cycles Kovács, Benedek Nagy, Zoltán Lóránt Combinatorics In this paper we continue the study of a natural generalization of Turán's forbidden subgraph problem and the Ruzsa-Szemerédi problem. Let $ex_F(n,G)$ denote the maximum number of edge-disjoint copies of a fixed simple graph $F$ that can be placed on an $n$-vertex ground set without forming a subgraph $G$ whose edges are from different $F$-copies. The case when both $F$ and $G$ are triangles essentially gives back the theorem of Ruzsa and Szemerédi. We extend their results to the case when $F$ and $G$ are arbitrary cliques by applying a number theoretic result due to Erdős, Frankl and Rödl. This extension in turn decides the order of magnitude for a large family of graph pairs, which will be subquadratic, but almost quadratic. Since the linear $r$-uniform hypergraph Turán problems to determine $ex_r^{lin}(n,G)$ form a class of the multicolor Turán problem, following the identity $ex_r^{lin}(n,G)=ex_{K_r}(n,G)$, our results determine the linear hypergraph Turán numbers of every graph of girth $3$ and for every $r$ up to a subpolynomial factor. Furthermore, when $G$ is a triangle, we settle the case $F=C_5$ and give bounds for the cases $F=C_{2k+1}$, $k\ge 3$ as well. |
| title | Multicolor Turán numbers II -- a generalization of the Ruzsa-Szemerédi theorem and new results on cliques and odd cycles |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2207.14572 |